Next, based on the \((O(1),\tilde{O}(k))\)  bicriteria solution, we perform non-adaptive residual sampling. This was shown to give a \((1+\epsilon , \tilde{O}(k^{3}/\epsilon ^2))\)  bicriteria solution in [1]} when an \(O(1)\)  approximate solution is used. Thus, we obtain a subspace \(\hat{S}\)  for which
\(\Vert A(I-P)(I-\mathbb {P}_{\hat{S}})\Vert _{1,2} \le (1+\epsilon )\textnormal {SubApx}_{k,1}(A(I-P)).\) 
